Muon Radiography allows to map the density of a volcanic cone. It is based on the measurement of the attenuation of the flux of muons of cosmic origin inside the matter. The MU-RAY project has developed an innovative detector designed for the muon radiography. The main features are the low electric power consumption, robustness and transportability, high spatial resolution and muon time of flight measurement. A 1 m^2 detector prototype has been constructed and has been employed in measurement campaigns at Mt. Vesuvius for approximately 1 month in spring 2013, and at the Puy de Dˆome, France. In this thesis we show the principles of muon radiography, the state of art of the detectors for muon radiography, the MU-RAY detector design and realization and its last upgrades and improvements.